“TAKE HOME” PORTION OF THE FINAL EXAM, DUE ON
EXAM DAY
(1) Let
R
d
be the (infinite) poset, whose ground set is the set of
d
tuples of
real numbers, and (
a
1
,...,a
d
)
≤
(
b
1
,...,b
d
) if and only if
a
i
≤
b
i
for all
i
= 1
,...,d
. Prove that the dimension of a poset
P
is equal to the least
d
such that
P
can be embedded
1
into
R
d
.
(2) Let
X
=
{
a,b,c,d,e,f
}
be the ground set of a poset.
Define the (strict,
i.e. irreflexive) relation to be
{
(
a,b
)
,
(
a,c
)
,
(
a,d
)
,
(
b,d
)
,
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 Fall '09
 WILDSTROM
 Real Numbers, Order theory, Rational number, poset, injective orderpreserving map

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